In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra. The theory of cyclic extensions of the field K when the characteristic of K does divide n is called Artin–Schreier theory.
Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.
In abstract algebra and number theory, Kummertheory provides a description of certain types of field extensions involving the adjunction of nth roots...
congruence Kummer series KummertheoryKummer's theorem, on prime-power divisors of binomial coefficients Kummer's function Kummer sum Kummer variety Kummer–Vandiver...
solvable extensions correspond to solvable groups. Theories such as Kummertheory and class field theory are predicated on the fundamental theorem. Given...
Thomas John Kummer (October 10, 1933 – August 9, 1969), known professionally as Jay Sebring, was an American celebrity hair stylist, and the founder of...
profinite groups. Kummertheory The Galois theory of taking nth roots, given enough roots of unity. It includes the general theory of quadratic extensions...
geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds. Class field theoryKummertheory Locally compact field Tamagawa...
integers.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws, that is,...
the application. In category theory, 1 is the terminal object of a category if there is a unique morphism. In number theory, 1 is the value of Legendre's...
elements. Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In order theory (and especially its...
Applications Galois theory Galois group Inverse Galois problem Kummertheory General Module (mathematics) Bimodule Annihilator (ring theory) Structure Submodule...
Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group G of a certain arithmetic variety X,...
application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study...
In algebraic geometry, a Kummer quartic surface, first studied by Ernst Kummer (1864), is an irreducible nodal surface of degree 4 in P 3 {\displaystyle...
unity, giving a non-abelian Galois group as semi-direct product. The Kummertheory gives a complete description of the abelian extension case, and the...
abelian varieties now includes Diophantine geometry along with class field theory, complex multiplication, local zeta-functions and L-functions. Paul Vojta...
theory of abelian varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory...
complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy. A Kummer sum is therefore a finite sum ∑ χ ( r ) e ( r / p ) =...
rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. Analysis on Lie groups and certain...
Daniel (2010). Fundamental Interactions: A memorial volume for Wolfgang Kummer. World Scientific. p. 351. ISBN 978-981-4277-83-9 – via Google Books. Tsybychev...
bassoon, oboe, cello and double bassoon. Kummer was taught by Neumeister for a year, then took training in music theory from the Cantor of Schleusingen, Gottlob...
hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solution to Kummer's differential equation...
Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummertheory. Let...
unity w ( P , Q ) ∈ μ n {\displaystyle w(P,Q)\in \mu _{n}} by means of Kummertheory, for any two points P , Q ∈ E ( K ) [ n ] {\displaystyle P,Q\in E(K)[n]}...
the work of man"). Kronecker was a student and life-long friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now...
reducibilities and other related notions. One of the major results was Kummer's Cardinality Theory which states that a set A is computable if and only if there...
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